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  <title>数学-高等数学 5 第13讲 多元函数微分学</title>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#13">第13讲 多元函数微分学</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1">1. 导数与微分</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_1">（1）偏导数的定义公式</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2">（2）二元函数微分的定义</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_1">2. 复合函数求导法</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_2">（1）链式求导法则</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_2">（2）全导数</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#3">（3）全微分形式不变性</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3_1">3. 隐函数求导法</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_3">（1）一个方程的情形</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_3">（2）方程组情形</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4-fxy">4. 二元函数 <script type="math/tex">f(x,y)</script> 极值</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5-fxyz0-zxy-zxy">5. 多元函数隐函数 <script type="math/tex"> F(x,y,z)=0 </script> 确定 <script type="math/tex">z(x,y)</script>，求 <script type="math/tex">z(x,y)</script> 的极值</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#6">6. 条件极值（拉格朗日数乘法）</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#fxy">【二元】函数 <script type="math/tex">f(x,y)</script> 在条件</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#fxyz">【三元】函数 <script type="math/tex">f(x,y,z)</script> 在条件</a>
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  <a class="pure-menu-link nav5" onclick="animateByNav()" href="#_1">例题</a>
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  <h1 id="数学-高等数学 5 第13讲 多元函数微分学" class="content-subhead">数学-高等数学 5 第13讲 多元函数微分学</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
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    <h2 id="13">第13讲 多元函数微分学</h2>
<h3 id="1">1. 导数与微分</h3>
<p>
<script type="math/tex; mode=display">
偏导数存在 \Rightarrow 可微
\begin{cases}
\Rightarrow 偏导数存在 &（某方向双侧）\\[2ex]
\Rightarrow 连续 \Rightarrow 极限存在 &（全方向） \\[2ex]
\Rightarrow 方向导数存在 &（某方向单侧）
\end{cases}
</script>
</p>
<h4 id="1_1">（1）偏导数的定义公式</h4>
<p>
<script type="math/tex; mode=display">
f'_x(x,y) = \lim_{\Delta x\to0}\cfrac{f(x_0 + \Delta x, y_0) - f(x_0, y_0)}{\Delta x}
</script>
</p>
<h4 id="2">（2）二元函数微分的定义</h4>
<p>设函数 <script type="math/tex">z=f(x,y)</script> 在点 <script type="math/tex">(x_0,y_0)</script> 的某领域内有定义，且 <script type="math/tex">(x_0+\Delta x,y_0+\Delta y)</script> 在该领域内，对于 <strong>全增量</strong><br />
<script type="math/tex; mode=display">
\Delta z = f(x_0+\Delta x, y_0+\Delta y)-f(x_0,y_0)
</script>
<br />
若存在与 <script type="math/tex">\Delta x, \Delta y</script>
<strong>无关</strong>，而仅与 <script type="math/tex">x,y</script>
<strong>有关</strong> 的常数 <script type="math/tex">A,B</script> 使得<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\Delta z &= A\Delta x + B\Delta y+o(\sqrt{(\Delta x)^2+(\Delta y)^2}) \\[1ex]
&= 线性增量 + 高阶无穷小量
\end{split}\end{equation}
</script>
<br />
则称 <script type="math/tex">f(x,y)</script> 在 <script type="math/tex">(x_0,y_0)</script> 处 <strong>可微</strong>，并称 <script type="math/tex">A\Delta x + B\Delta y</script> 为 <script type="math/tex">f(x,y)</script> 在点 <script type="math/tex">(x_0,y_0)</script> 处的 <strong>全微分</strong>，记作<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
dz\bigg|_{(x,y)=(x_0,y_0)} &= A\Delta x + B\Delta y\\[1ex]
&= Adx + Bdy\\[1em]
偏导数：f_x'(x_0,y_0) = \cfrac{dz}{dx}\bigg|_{(x,y)=(x_0,y_0)} &= A \\[1ex]
偏导数：f_y'(x_0,y_0) = \cfrac{dz}{dy}\bigg|_{(x,y)=(x_0,y_0)} &= B
\end{split}\end{equation}
</script>
<br />
<script type="math/tex">z=f(x,y)</script> 在点 <script type="math/tex">(x_0,y_0)</script> 处 <strong>可微的条件</strong><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
 &\lim_{\Delta x \to 0,\Delta y \to 0}\cfrac{\Delta z - (A\Delta x + B\Delta y)}
 {\sqrt{(\Delta x)^2+(\Delta y)^2}}=&0
\end{split}\end{equation}
</script>
</p>
<p>偏导数连续的条件（<strong>注意是从各个方向靠近</strong>）<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\lim_{(x,y)\to(x_0,y_0)}f_x'(x,y)=f_x'(x_0,y_0) \\[1ex]
\lim_{(x,y)\to(x_0,y_0)}f_y'(x,y)=f_y'(x_0,y_0)
\end{split}\end{equation}
</script>
</p>
<h3 id="2_1">2. 复合函数求导法</h3>
<h4 id="1_2">（1）链式求导法则</h4>
<p>设 <script type="math/tex"> z=z(u,v),u=u(x,y),v=v(x,y) </script>
<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\begin{pmatrix}
\cfrac{\partial z}{\partial x} &\cfrac{\partial z}{\partial y} \\
\end{pmatrix} &=
\begin{pmatrix}
\cfrac{\partial z}{\partial u} &\cfrac{\partial z}{\partial v} \\
\end{pmatrix}
\begin{pmatrix}
\cfrac{\partial u}{\partial x} &\cfrac{\partial u}{\partial y} \\
\cfrac{\partial v}{\partial x} &\cfrac{\partial v}{\partial y} \\
\end{pmatrix} \\[1ex]
\cfrac{\partial z}{\partial x} &= \cfrac{\partial z}{\partial u}\cfrac{\partial u}{\partial x}
+ \cfrac{\partial z}{\partial v}\cfrac{\partial v}{\partial x} \\[1ex]
\cfrac{\partial z}{\partial y} &= \cfrac{\partial z}{\partial u}\cfrac{\partial u}{\partial y}
+ \cfrac{\partial z}{\partial v}\cfrac{\partial v}{\partial y}
\end{split}\end{equation}
</script>
</p>
<h4 id="2_2">（2）全导数</h4>
<p>设 <script type="math/tex"> z=z(u,v),u=u(x),v=v(x) </script> ，即 <script type="math/tex"> z </script> 最终是 <script type="math/tex"> x </script> 的函数，则 <script type="math/tex"> \cfrac{dz}{dx} </script> 叫 <strong>全导数</strong><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{dz}{dx} &=
\begin{pmatrix}
\cfrac{\partial z}{\partial u} &\cfrac{\partial z}{\partial v} \\
\end{pmatrix}
\begin{pmatrix}
\cfrac{\partial u}{\partial x} \\ 
\cfrac{\partial v}{\partial x}
\end{pmatrix} \\[1ex]
&= \cfrac{\partial z}{\partial u}\cfrac{\partial u}{\partial x}
+ \cfrac{\partial z}{\partial v}\cfrac{\partial v}{\partial x}
\end{split}\end{equation}
</script>
</p>
<h4 id="3">（3）全微分形式不变性</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
dz &= \cfrac{\partial z}{\partial x}dx + \cfrac{\partial z}{\partial y}dy\\
   &= \cfrac{\partial z}{\partial u}du + \cfrac{\partial z}{\partial v}dv
\end{split}\end{equation}
</script>
</p>
<h3 id="3_1">3. 隐函数求导法</h3>
<h4 id="1_3">（1）一个方程的情形</h4>
<p>将 <script type="math/tex">z=z(x,y)</script> 转化为 <script type="math/tex">F(x,y,z)=0</script> 求偏导<br />
<script type="math/tex; mode=display">
\cfrac{\partial z}{\partial x} = -\cfrac{F'_x}{F'_z}\\
\cfrac{\partial z}{\partial y} = -\cfrac{F'_y}{F'_z}
</script>
</p>
<h4 id="2_3">（2）方程组情形</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\begin{cases}
F(x,u,v)=0 \\[1ex]
G(x,u,v)=0
\end{cases}
，当满足
\left|\begin{array}{c c}
\cfrac{\partial F}{\partial u} & \cfrac{\partial F}{\partial v} \\
\cfrac{\partial G}{\partial u} & \cfrac{\partial G}{\partial v} \\
\end{array}\right|
=\cfrac{\partial(F,G)}{\partial(u,v)}\neq0时，可以确定
\begin{cases}
u=u(x) \\[1ex]
v=v(x)
\end{cases}
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\cfrac{du}{dx}=-\cfrac{\cfrac{\partial(F,G)}{\partial(x,v)}}{\cfrac{\partial(F,G)}{\partial(u,v)}} 
\ \ \ \ \ \ \ \ \ 
\cfrac{dv}{dx}=-\cfrac{\cfrac{\partial(F,G)}{\partial(u,x)}}{\cfrac{\partial(F,G)}{\partial(u,v)}}
</script>
</p>
<blockquote class="content-quote">
<p>推导过程：<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\begin{cases}
\cfrac{\partial F}{\partial x} + \cfrac{\partial F}{\partial u}\cfrac{du}{dx} + \cfrac{\partial F}{\partial v}\cfrac{dv}{dx} = 0 \\[1ex]
\cfrac{\partial G}{\partial x} + \cfrac{\partial G}{\partial u}\cfrac{du}{dx} + \cfrac{\partial G}{\partial v}\cfrac{dv}{dx} = 0 \\[1ex]
\end{cases} \\[2ex]
⟹
&\begin{cases}
\cfrac{\partial F}{\partial u}\cfrac{du}{dx} + \cfrac{\partial F}{\partial v}\cfrac{dv}{dx} = -\cfrac{\partial F}{\partial x} \\[1ex]
\cfrac{\partial G}{\partial u}\cfrac{du}{dx} + \cfrac{\partial G}{\partial v}\cfrac{dv}{dx} = -\cfrac{\partial G}{\partial x} \\[1ex]
\end{cases} \\[2ex]
⟹
&\begin{cases}
a_{11}\cfrac{du}{dx} + a_{12}\cfrac{dv}{dx} = b_1 \\[1ex]
a_{21}\cfrac{du}{dx} + a_{22}\cfrac{dv}{dx} = b_2 \\[1ex]
\end{cases}
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
(克拉默法则)
\ \ \ \ \ \ \ \ \ 
\cfrac{du}{dx}=
-\cfrac
{\left|\begin{array}{c c}
b_1 & a_{12} \\
b_2 & a_{22} \\
\end{array}\right|}
{\left|\begin{array}{c c}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{array}\right|}
\ \ \ \ \ \ \ \ \ 
\cfrac{dv}{dx}=
-\cfrac
{\left|\begin{array}{c c}
a_{11} & b_1 \\
a_{21} & b_2 \\
\end{array}\right|}
{\left|\begin{array}{c c}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{array}\right|}
</script>
</p>
</blockquote>
<h3 id="4-fxy">4. 二元函数 <script type="math/tex">f(x,y)</script> 极值</h3>
<p>先求一阶偏导数 <br />
<script type="math/tex; mode=display">
f'_x(x,y)=0 \\[1ex]
f'_y(x,y)=0
</script>
</p>
<p>再求二阶偏导数</p>
<p>
<script type="math/tex; mode=display">
A = f''_{xx}(x_0, y_0) \\[1ex]
B = f''_{xy}(x_0, y_0) \\[1ex]
C = f''_{yy}(x_0, y_0)
</script>
</p>
<p>
<script type="math/tex"> (x_0, y_0) </script>  为一阶导数  <script type="math/tex"> f_x(x,y) = 0，f_y(x,y) = 0 </script>  的点</p>
<ol>
<li>
<script type="math/tex"> AC-B^2 \gt 0 </script>  且  <script type="math/tex"> A \gt 0 </script>  极小值点</li>
<li>
<script type="math/tex"> AC-B^2 \gt 0 </script>  且  <script type="math/tex"> A \lt 0 </script>  极大值点</li>
<li>
<script type="math/tex"> AC-B^2 \lt 0 </script>  非极值点（鞍点）</li>
<li>
<script type="math/tex"> AC-B^2 = 0 </script>  不确定</li>
</ol>
<blockquote class="content-quote">
<p>二元函数取极值</p>
<p><strong>必要条件</strong>：各分量的偏导数为，<script type="math/tex">f_x'(x_0,y_0)=0,f_y'(x_0,y_0)=0</script>
</p>
<p><strong>充分条件</strong>：二阶偏导数组成的矩阵<br />
<script type="math/tex; mode=display">
\left[\begin{array}{c c}
A & B \\
B & C
\end{array}\right]
</script>
<br />
为 <strong>正定矩阵</strong> 或 <strong>负定矩阵</strong></p>
<ul>
<li>正定矩阵：所有顺序主子式都 <strong>大于</strong> 0</li>
<li>此时取到极大值，<script type="math/tex">AC-B^2>0,A>0</script>
</li>
<li>负定矩阵：偶数阶顺序主子式都 <strong>大于</strong> 0，奇数阶顺序主子式都 <strong>小于</strong> 0</li>
<li>此时取到极小值，<script type="math/tex">AC-B^2>0,A<0</script>
</li>
</ul>
</blockquote>
<h3 id="5-fxyz0-zxy-zxy">5. 多元函数隐函数 <script type="math/tex"> F(x,y,z)=0 </script> 确定 <script type="math/tex">z(x,y)</script>，求 <script type="math/tex">z(x,y)</script> 的极值</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\begin{cases}
F_x'+F_z'z_x'=0 \\[6ex]
F_y'+F_z'z_y'=0
\end{cases}
&\Rightarrow
\begin{cases}
z_x' = -\cfrac{F_x'}{F_z'} = 0 \\[2ex]
z_y' = -\cfrac{F_y'}{F_z'} = 0
\end{cases}
\Rightarrow \begin{cases}
F_x' = 0 \\[6ex]
F_y' = 0
\end{cases} \\[4ex]
&\Rightarrow P_0(x_0,y_0)、P_1(x_1,y_1)，...\ \ \ \ 得到极值点\\[2em]
A = z_{xx}'' 
&= \cfrac{d(z_x')}{dx} \\[1ex]
&= \cfrac{d(-\cfrac{F_x'}{F_z'})}{dx} \\[1ex]
&= -\cfrac{(F_{xx}''+F_{xz}''z_x')F_z'-F_x'(F_{zx}''+F_{zz}''z_x')}{(F_z')^2} \\[1ex]
&= -\cfrac{F_{xx}''}{F_z'} \\[2ex]
B = z_{xy}''
&= -\cfrac{F_{xy}''}{F_z'} \\[2ex]
C = z_{xy}''
&= -\cfrac{F_{yy}''}{F_z'} \ \ \ \ 判断极大值极小值
\end{split}\end{equation}
</script>
</p>
<h3 id="6">6. 条件极值（拉格朗日数乘法）</h3>
<h4 id="fxy">【二元】函数 <script type="math/tex">f(x,y)</script> 在条件</h4>
<p>
<script type="math/tex; mode=display">
\varphi(x,y) =0
</script>
</p>
<p>下取得极值的必要条件：</p>
<p>
<script type="math/tex; mode=display">
F(x,y,\lambda) = f(x,y)+\lambda\varphi(x,y) \\[2em]
\begin{cases}
F'_x(x,y,\lambda) = f'_x(x,y) + \lambda \varphi'_x(x,y) = 0 \\[2ex]
F'_y(x,y,\lambda) = f'_y(x,y) + \lambda \varphi'_y(x,y) = 0 \\[2ex]
F'_\lambda(x,y,\lambda) = \varphi(x,y)= 0 \\
\end{cases}
</script>
</p>
<h4 id="fxyz">【三元】函数 <script type="math/tex">f(x,y,z)</script> 在条件</h4>
<p>
<script type="math/tex; mode=display">
\begin{cases}
\varphi(x,y,z)=0 \\[2ex]
\psi(x,y,z)=0
\end{cases}
</script>
</p>
<p>下取得极值的必要条件：</p>
<p>
<script type="math/tex; mode=display">
F(x,y,z,\lambda,\mu)=f(x,y,z)+\lambda\varphi(x,y,z)+\mu\psi(x,y,z) \\[2em]
\begin{cases}
F'_x(x,y,z,\lambda,\mu) = f'_x(x,y,z) + \lambda \varphi'_x(x,y,z) + \mu \psi'_x(x,y,z) = 0 \\[2ex]
F'_y(x,y,z,\lambda,\mu) = f'_y(x,y,z) + \lambda \varphi'_y(x,y,z) + \mu \psi'_y(x,y,z) = 0 \\[2ex]
F'_z(x,y,z,\lambda,\mu) = f'_z(x,y,z) + \lambda \varphi'_z(x,y,z) + \mu \psi'_z(x,y,z) = 0 \\[2ex]
F'_\lambda(x,y,z,\lambda,\mu) = \varphi(x,y,z) = 0 \\[2ex]
F'_\mu(x,y,z,\lambda,\mu) = \psi(x,y,z)= 0 \\
\end{cases}
</script>
</p>
<h5 id="_1">例题</h5>
<blockquote class="content-quote">
<p>函数 <script type="math/tex">f(x,y)</script> 在条件 <script type="math/tex">\phi(x,y)=0</script> 下取得极值的必要条件，设 <script type="math/tex">(x_0,y_0)</script> 处为极值点，则<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\phi(x_0,y_0) &= 0 \\[2ex]
df &= f'_x(x,y)dx + f'_y(x,y)dy  \\[2ex]
\Rightarrow &\begin{cases}
\cfrac{df}{dx}_{(x,y)=(x_0,y_0)} = f'_x(x_0,y_0) + f'_y(x_0,y_0)\cfrac{dy}{dx}_{(x,y)=(x_0,y_0)} = 0 \\[2ex]
\cfrac{df}{dy}_{(x,y)=(x_0,y_0)} = f'_y(x_0,y_0) + f'_x(x_0,y_0)\cfrac{dx}{dy}_{(x,y)=(x_0,y_0)} = 0 \\[2ex]
\cfrac{dy}{dx}_{(x,y)=(x_0,y_0)} = -\cfrac{\phi'_x(x_0,y_0)}{\phi'_y(x_0,y_0)} \\[2ex]
\cfrac{dx}{dy}_{(x,y)=(x_0,y_0)} = -\cfrac{\phi'_y(x_0,y_0)}{\phi'_x(x_0,y_0)}
\end{cases} \\[2em]
\Rightarrow &\begin{cases}
\cfrac{df}{dx}_{(x,y)=(x_0,y_0)} =& f'_x(x_0,y_0) -\cfrac{f'_y(x_0,y_0)}{\phi'_y(x_0,y_0)}\phi'_x(x_0,y_0) = 0 \\[2ex]
&f'_x(x_0,y_0)-\cfrac{f'_x(x_0,y_0)}{\phi'_x(x_0,y_0)}\phi'_x(x_0,y_0) = 0 \\[2ex]
\cfrac{df}{dy}_{(x,y)=(x_0,y_0)} =& f'_y(x_0,y_0) -\cfrac{f'_x(x_0,y_0)}{\phi'_x(x_0,y_0)}\phi'_y(x_0,y_0) = 0 \\[2ex]
&f'_y(x_0,y_0)-\cfrac{f'_y(x_0,y_0)}{\phi'_y(x_0,y_0)}\phi'_y(x_0,y_0) = 0 \\[2ex]
\end{cases} \\[2em]
综上&\begin{cases}
f'_x(x_0,y_0)+\lambda\phi'_x(x_0,y_0) = 0 \\[2ex]
f'_y(x_0,y_0)+\lambda\phi'_y(x_0,y_0) = 0 \\[2ex]
\phi(x_0,y_0) = 0 \\[2ex]
\end{cases}
，且\lambda = -\cfrac{f'_x(x_0,y_0)}{\phi'_x(x_0,y_0)} = -\cfrac{f'_y(x_0,y_0)}{\phi'_y(x_0,y_0)}
\end{split}\end{equation}
</script>
<br />
恰好与下式相等</p>
</blockquote>
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